A physically motivated parametric model for
compact representation of room impulse
responses based on orthonormal basis functions
Giacomo Vairetti, Enzo De Sena, Toon van Waterschoot∗ , Marc Moonen
Dept. of Electrical Engineering (ESAT), STADIUS Center for Dynamical Systems, Signal Processing
and Data Analytics, KU Leuven, Kasteelpark Arenberg 10, 3001 Leuven, Belgium.
Michael Catrysse
Televic N.V., Leo Bekaertlaan 1, 8870 Izegem, Belgium.
Neofytos Kaplanis†
Bang & Olufsen A/S, Peter Bangs Vej 15, 7600 Struer, Denmark.
Søren Holdt Jensen
Dept. of Electronic Systems, Aalborg University, Niels Jernes Vej 12, 9220 Aalborg, Denmark.
Summary
A Room Impulse Response (RIR) shows a complex time-frequency structure, due to the presence
of sound reections and room resonances at low frequencies. Many acoustic signal enhancement
applications, such as acoustic feedback cancellation, dereverberation and room equalization, require
simple yet accurate models to represent a RIR. Parametric modeling of room acoustics attempts
at approximating the Room Transfer Function (RTF), for given positions of source and receiver
inside a room, by means of rational functions in the
z -domain
that can be implemented through
digital lters. However, conventional parametric models, such as all-zero and pole-zero models, have
some limitations. In this paper, a particular xed-pole Innite Impulse Response (IIR) lter based
on Orthonormal Basis Functions (OBFs) is used as an alternative, motivated by its analogy to the
physical denition of the RIR as a Green's function of the acoustic wave equation. An accurate
estimation of the model parameters allows arbitrary allocation of the spectral resolution, so that the
room resonances can be described well and a compact representation of a target RIR can be achieved.
The model parameters can be estimated by a scalable matching pursuit algorithm called OBF-MP,
which selects the most prominent resonance at each iteration. A modied version of the algorithm,
called OBF-GMP (Group Matching Pursuit), is introduced for the estimation of a common set of
poles from multiple RIRs measured at dierent positions inside a room. A new database of RIRs
measured in a rectangular room using a subwoofer is also presented. Simulation results using this
database show that, in comparison to OBF-MP, the OBF-GMP signicantly reduces the number of
parameters necessary to represent the RIRs.
PACS no. xx.xx.Nn, xx.xx.Nn
1.
ral structure of sound reections. Parametric models
are used in all those acoustic signal enhancement applications that require the RIR to be represented in a
simple yet accurate way. Examples of these applications are acoustic feedback cancellation, dereverberation, and room equalization. In parametric modeling,
a Room Transfer Function (RTF), corresponding to a
Green's function of the acoustic wave equation for specic positions of the loudspeaker and the microphone
inside a room, is represented by means of a rational
function in the z -domain and implemented through
digital lters. This rational function can be written in
Introduction
A Room Impulse Response (RIR) shows a complex
time-frequency structure, due to the presence of room
resonances at low frequencies and the intricate tempo-
(c) European Acoustics Association
also aliated with the Dept. of Electrical Engineering (ESATETC), AdvISe Lab, KU Leuven, Kleinhoefstraat 4, 2440 Geel,
Belgium.
† also aliated with the Dept. of Electronic Systems, Aalborg
University, Niels Jernes Vej 12, 9220 Aalborg, Denmark.
∗
Copyright© (2015) by EAA-NAG-ABAV, ISSN 2226-5147
All rights reserved
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terms of zeros and poles by computing the complexvalued roots of the numerator and denominator polynomials, respectively. However, conventional parametric models, such as all-zero and pole-zero models,
present some limitations. The all-zero model [1] uses a
Finite Impulse Response (FIR) lter to approximate
the sampled RIR, with the number of parameters corresponding to the sample index at which the RIR is
truncated. A zero approximation error is obtained up
to the truncation index, but a large number of parameters is required in order to capture the resonant
characteristics of the room, especially when the reverberation time is high. Moreover, the parameter values
are strongly dependent on the source and receiver positions. All-pole and pole-zero models are used in an
attempt to overcome these limitations. These models
use pairs of complex-conjugate poles to represent resonances in the RTF. This enables to reduce the number
of parameters and to obtain parameter values less sensitive to changes in the source and receiver positions.
However, a stable all-pole model cannot represent true
delays nor the non-minimum-phase characteristics of
the RTF [1]. Pole-zero models [2], on the other hand,
represent resonances and damping constants by the
poles of the RTF, and anti-resonances and time delays
by its zeros. The Common-Acoustical-Pole and Zero
(CAPZ) model [3] exploits the fact that room resonances are independent of the position of the source
and receiver, but are rather a characteristic of the
room itself. As the name suggests, the RTFs measured
at dierent positions in the room are parametrized
by a common set of poles, while dierences between
these responses are described by dierent sets of zeros. In this way, a more compact representation of a
group of RIRs is obtained. However, since the poles
appear nonlinearly in the pole-zero model, no closedform solution to the parameter estimation problem exists, thus requiring nonlinear optimization techniques,
possibly leading to instability or convergence to local
minima.
the sense that poles can be distributed arbitrarily inside the unit circle of the z -plane, thus giving freedom
in the allocation of the spectral resolution. However,
since OBF models are nonlinear in the pole parameters, estimating the poles that provide a good approximation of a given RIR is a nonlinear problem. Nonlinear optimization techniques have been proposed for
the optimization of the poles in dierent applications,
such as acoustic echo cancellation [8] and loudspeaker
and room modeling [9]. In [10], the nonlinear problem was avoided by iteratively selecting poles from a
discrete grid using a scalable matching pursuit algorithm, called OBF-MP.
This paper introduces a modied version of the
OBF-MP that aims at estimating a set of poles common to multiple RIRs measured at dierent positions
in the same room. It is shown that the modied algorithm, termed here OBF-GMP, approximates the set
of RIRs more accurately, compared to the case when
the poles are estimated individually for each RIR or
when the all-zero model is used, with the same total number of parameters. Simulations have been performed on a database of RIRs measured at dierent
positions inside a rectangular room, with a subwoofer
source.
The paper is structured as follows. OBF models are
described in Section 2. In Section 3, the OBF-GMP
algorithm is introduced. Section 4 describes the RIR
database measurements. Simulation results are presented in Section 5. Section 6 concludes the paper
and indicates possible directions for future work.
2.
OBF models
OBF models dene an IIR lter structure that allows to incorporate information about the resonant
and damping behavior of a system. Although these
models are widely used in system identication and
control applications, only a few examples of their use
in room acoustics modeling are known [8, 9, 10, 11].
Under fairly realistic assumptions, a room can be considered as a causal, stable, and linear system, which
is characterized by a number of room resonances, or
modes. A mode is represented in the z -domain by a
second-order resonator dened by a pair of complexconjugate poles {pi , p∗i }, with transfer function
An alternative to conventional parametric models
is provided by a particular family of models based on
orthonormal basis functions.Orthonormal Basis Function (OBF) models [4, 5, 6] dene a xed-pole IIR lter, which is an orthonormalized parallel connection
of second-order resonators, whose impulse responses
represent damped sinusoids. Then, the RIR approximation is built as a linear superposition of a nite
number of exponentially decaying sinusoids, whose
frequency and decay rate is determined by the position of the poles inside the unit circle. The analogy
with the denition of the RTF is clear. Each term
of the Green's function corresponds to a resonator
whose impulse response is a sinusoid, oscillating at
a particular resonance frequency and damped with a
particular damping constant [7]. OBF models possess
many other desirable properties, such as orthogonality
and stability. These models are also very exible, in
Pi (z) =
1
(1 − pi z )(1 − p∗i z −1 )
−1
(1)
with ∗ indicating complex conjugation. The impulse
response corresponding to the transfer function in (1)
is an exponentially decaying sinusoid, sampled at frequency fs , oscillating at the resonant frequency ωi
and decaying exponentially accordingly to the damping constant ζi , which also determines the bandwidth
of the resonance. Thus, the angle ϑi = ωi /fs and the
radius ρi = e−ζi /fs of the pole pair {pi , p∗i } = ρi e±jϑi
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δ(n)
z −d
A1 (z)
P1 (z)
N1+ (z)
ϕ+
1 (n)
G. Vairetti et al.: A...
P2 (z)
N1− (z)
ϕ−
1 (n)
θ1−
θ1+
Pm (z)
N2− (z)
N2+ (z)
ϕ+
2 (n)
Secondly, the OBFs form a complete set in the
Hardy space
P on the unit disc under the mild assumption that ∞
i=1 (1 − |pi |) = ∞ [6]. Thus, any stable
linear lter can be approximated with arbitrary accuracy by a linear combination of a certain nite number
of OBFs, so that OBF models can achieve an accurate
approximation of a RIR with a reduced number of parameters, compared to conventional models.
A third important property is that OBF models are linear in the coecients θi± (cfr. Figure 1).
The approximation of a RIR h(n) as a combination of exponentially decaying sinusoidal responses
ϕ±
i (n, pi ) consists in estimating the pole parameters pi = {pi , p∗i } and the linear coecients θi =
{θi+ , θi− } (where i = 1, . . . , m) that minimize the distance between h(n) and the approximated response
ĥ(n, p, θ) for n = 1, . . . , N (with n = t/fs the discrete
time variable). Given the sets p = {p1 , . . . , pm } and
θ = {θ1 , . . . , θm }, the output ĥ(n, p, θ) of an OBF
model for an impulse input signal δ(n) is the linear
combination of the 2m basis functions ϕ±
i (n, pi ) =
Qi−1
[Ni± (z)Pi (z) j=1 Aj (z)]δ(n),
Am−1 (z)
ϕ−
2 (n)
θ2+
+
Nm
(z)
ϕ+
m (n)
θ2−
−
Nm
(z)
ϕ−
m (n)
+
θm
−
θm
ĥ(n, p, θ)
Figure 1. The generalized OBF model structure for
m
pairs of complex-conjugate poles.
determine the shape of the resonance. For instance,
poles close to the unit circle correspond to room
modes with long decay time and high Q-value. Multiple resonances can be represented by a parallel connection of second-order resonators. The resulting lter structure, also called parallel second-order lter
[12], originates from a partial fraction expansion of
the pole-zero RTF. The terms corresponding to a pair
of complex-conjugate poles {pi , p∗i } are combined to
form a second-order IIR lter with real-valued coecients and transfer function as in (1), which produces
a pair of real-valued responses, one being the onesample delayed version of the other. The output of
this lter structure is then a linear combination of
pairs of damped sinusoids, which are used as basis
functions in a linear-in-the-parameters model.
OBF models originate from the orthonormalization
of the parallel second-order lter, where orthogonality between any two consecutive basis functions is enforced by a second-order all-pass lter,
Ai (z) =
(z −1 − pi )(z −1 − p∗i )
,
(1 − pi z −1 )(1 − p∗i z −1 )
ĥ(n, p, θ) =
i=1
r
= |1 ± pi |
1 − |pi |2 −1
(z ∓ 1).
2
+
ϕ+
i (n, pi )θi +
m
X
−
ϕ−
(4)
i (n, pi )θi
i=1
or ĥ(n, p, θ) = ϕ(n, p)T θ, where ϕ(n, p) is a vector
containing the impulse responses ϕ±
i (n, pi ) at time
n. For a xed set of poles p, the problem of estimating the coecients in θ becomes linear and can
be solved in closed form using linear regression. By
stacking all the vectors ϕ(n, p) for n = 1, . . . , N in a
matrix Φ(p), the optimal values in the least-squares
sense for a given data set {h} = {h(n)}N
n=1 are obtained as θ̂ = Φ(p)T h, given that the orthonormality
of the basis functions implies Φ(p)T Φ(p) = IN .
The problem then reduces to the estimation of
the poles, requiring in principle nonlinear estimation
methods. The only known nonlinear method for the
estimation of the poles for RIR approximation using
OBF is the one proposed in [9]. In [11], the nonlinear
problem was avoided by selecting poles from a large
grid of candidate poles using a convex optimization
method. An iterative greedy algorithm, called OBFMP, was introduced in [10], which is scalable and numerically well-conditioned. In the following section,
the OBF-MP algorithm is modied in order to jointly
estimate a set of poles, common to multiple RIRs measured in the same room.
(2)
in which the zeros in 1/pi and 1/p∗i ensure that the basis functions dened by {pi+1 , p∗i+1 } are orthogonal to
those generated by {pi , p∗i }. The two basis functions
of each pole pair are normalized and made orthogonal
to each other by a pair of orthonormalization lters
Ni± (z). The general lter structure of an OBF model
is shown in Figure 1 for m pairs of complex-conjugate
poles. Dierent choices are available for the orthonormalization lters, as explained in [6]. In this work, the
so-called Kautz model [13] is used, where
Ni± (z)
m
X
(3)
3.
OBF models present many interesting properties.
Firstly, as opposed to pole-zero models, poles can
be positioned anywhere inside the unit circle without problems of numerical ill-conditioning. This also
allows to allocate a higher spectral resolution in the
frequency range of interest.
OBF-GMP algorithm
The OBF-MP [10] is a matching pursuit algorithm
which at each iteration selects the predictors, i.e.
the pair of basis functions, that are most correlated
with the current residual part of the RIR. The candidate predictors are generated based on a grid pg =
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1
1
0.5
0.5
0
0
−0.5
−1
−0.5
-1 -0.5 5 0.5 1
Real part
-1 -0.5 5 0.5 1
Real part
−1
Imaginary part
Imaginary part
G. Vairetti et al.: A...
ϕ−
i
r
αir−
αir
ϕ+
i
αir+
Figure 2. Pole grids using 500 poles, with 50 values for the
angle [1, fs/2 −1] Hz and 10 values for the radius [0.75,0.99].
Figure 3. Graphical interpretation of the correlation be-
(left) Logarithmic angles. (right) Logarithmic radii.
complex-conjugate poles
tween the residual vector
and the
{pi , p∗i }.
predictors of a pair of
Algorithm 1 OBF-GMP algorithm
{p1 , . . . , pG } of poles distributed inside the unit cir-
1: pg = {p1 , . . . , pG }
. Dene poles in the pole grid
2: nA = 0, k = 0 . n : # of selected predictors, k: iterations
3: E0 = Υ, Υ̂0 = 0,
. Initialize signal vectors
4: while nA < M do
. M : desired model order
5: Build Φk (pg ) . Φ : matrix of candidate predictors ϕ
6: j = arg maxi PRr=1 |αir | . Max. correlation with E
7: nA = nA + 2
. Update # of selected predictors
+ −
+ − T
r
8: ˆk = [ϕj ϕj ][αj αj ]
. Update approx. residual
1
R
9: Êk = [ˆk , . . . , ˆk ] . Current approx. residual matrix
10: Υ̂k+1 = Υ̂k + Êk . Update target approx. matrix
11: Ek+1 = Ek − Êk . Update current residual matrix
12: k = k + 1
13: end while
cle. The distribution of the poles in the grid is arbitrary and can be dictated by the desire of a higher
spectral resolution in the frequency range of interest
or by other considerations based on prior knowledge
about the acoustics of the room. Two examples are
given in Figure 2. The left example shows a pole grid
with angles distributed logarithmically, which yields a
higher resolution at low frequencies. The right example shows a pole grid with radii distributed logarithmically, which yields a higher resolution close to the unit
circle. At each iteration, the matrix Φk (pg ) is built
with the basis functions computed for each pole in
the grid pg and orthogonalized to the basis functions
added in previous iterations. The OBF-MP algorithm
is scalable. In fact, since the resulting lter structure
is orthogonal by construction, the linear coecients
do not have to be recomputed at each iteration. As
a consequence, the model order does not have to be
determined beforehand and more poles can be added
just by running extra iterations. At each iteration,
the approximation error is reduced and the algorithm
can be stopped when the desired degree of accuracy
is obtained. Orthogonality also implies that the linear coecients correspond to the correlation of the
basis functions with the RIR. It follows that no matrix inversion operation is involved in the algorithm,
avoiding any problem of numerical ill-conditioning.
Here, the OBF-MP algorithm is modied in order
to estimate a set of poles which is common to a set
of R RIRs measured in the same room. The modied algorithm, called OBF-GMP (Group Matching
Pursuit), is intended to reduce the number of parameters necessary to represent the RIRs by identifying the resonant characteristics of the room, common to all RIRs. The OBF-GMP algorithm, listed
below in details, works as follows. First, a grid pg of
G candidate poles has to be dened. Then, the R target RIRs hr = {hr (n)}N
n=1 are stacked in a matrix
Υ = [h1 , . . . , hR ], and the current residual matrix
E0 = [10 , . . . , R
0 ] is initialized as Υ. At each iteration k, OBF-GMP selects the pair of predictors in Φk
having maximum correlation with the residual signal
vectors in Ek . For a pair of complex-conjugate poles
{pi , p∗i }, the correlation αir with each residual vector
A
i
k
k
rk is chosen as the projection of rk on the plane de−
ned by predictors ϕ+
i and ϕi (see Figure 3), and is
given by
αir =
q
αir+ 2 + αir− 2 =
q
T
T
−
r 2
r 2
(ϕ+
i k ) + (ϕi k ) .
(5)
For each pair of complex-conjugate poles, the correlations with all the residual vectors in Ek are then
summed together, and the pole pair {pj , p∗j } selected
is the one with index
j = arg max
i
R
X
|αir |.
(6)
r=1
Given the orthogonal construction of the basis functions, each approximated residual vector is obtained
−
r
r T
as ˆrk = [ϕ+
and stacked in the maj ϕj ][αj + αj − ]
trix Êk , which is used for updating the current target approximation matrix Υ̂k = Êk + Υ̂k−1 , where
Υ̂ = [ĥ1 , . . . , ĥR ], and the residual signal matrix
Ek+1 = Ek − Êk . The algorithm terminates when the
desired number M of functions in the basis is reached
or when the approximation error falls below a desired
value.
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G. Vairetti et al.: A...
Table I. Room specications. Reverberation time calcu-
Table II. Microphone and speaker positions. Speaker Po-
lated with the backward integration method [18].
sition indicates the center of the transducer.
Dimensions
Reverberation Time
6.35
L × 4.09 W × 2.40 H (m)
Mic
X
1
2
3
4
5
6
T31.5Hz = 1.44s T40Hz = 0.69s
T50Hz = 0.74s T63Hz = 0.53s
T100Hz = 0.47s T125Hz = 0.62s
1.12
0.77
2.04
1.62
3.05
3.09
Y
1.56
4.04
4.47
5.32
3.06
5.07
Z
Src
1.50
1.80
0.90
0.60
1.50
1.00
1
2
3
4
X
3.84
2.90
3.63
2.35
Y
3.84
0.80
5.83
4.55
Z
0.53
0.53
0.53
1.13
hNMSE (dB)
0
Figure 4. Sketch of the room at B&O headquarters, Struer,
RIR database
0
200
400
C/R
600
800
Figure 5. The average NMSE w.r.t. the number of model
parameters
The RIR measurements used in the simulations were
performed in an unoccupied, standard domestic listening room, sketched in Figure 4, based at Bang
& Olufsen headquarters in Struer, Denmark. When
furnished, the room complies with IEC 60268-13
[14] with RT(500 Hz−1 KHz) = 0.35 s. The room comprises of wooden oor, wooden false ceiling lled with
Rockwool® , and lightly plastered painted walls with
high-frequency absorbing panels on the side walls.
During the measurements, the room was emptied, except for 8 acoustic wooden panels (0.5×0.5×0.025 m)
and 2 Helmholtz absorbers tuned at 200-300 Hz. The
room dimensions and the values of the reverberation time are given in Table I. RIR measurements
were obtained using the logarithmic sine-sweep technique [15] with a sampling rate of 48 kHz. The sweeps
were recorded with a B&K 4939 1/4" microphone
and B&K 2669 preamplier, connected to an audio interface (RME UCX) and a laptop computer.
Recordings of 3 s sine sweeps (0.1 Hz-1 KHz) produced
by a custom Genelec 1094A subwoofer (12-150 Hz,
±3 dB) were completed for 24 source-microphone positions (see Table II), in conformity with the guidelines in ISO 3382-1,2 [16, 17] for precision measurements. The RIR database measurements are available for download at http://www.dreams-itn.eu/
index.php/dissemination/downloads/subrir.
5.
−40
−60
Denmark.
4.
−20
C
per RIR. The average NMSE in (7) for
the entire time-response. All-Zero model ( ), OBF-GMP
model ( ), and OBF-MP model ( ).
R RIRs for a given number of parameters. For the allzero modeling, the number of parameters is R times
the number of taps used in the FIR lter for each response. For the OBF models (see Figure 1), the number of parameters is the number of poles m plus the
number of linear coecients 2m, which sum up to 3m
coecients. When estimating m poles individually for
each RIR with OBF-MP, the total number of parameters is 3mR. In case m poles are estimated jointly for
all RIRs with OBF-GMP, only one common set of m
poles is necessary, and the total number of parameters
becomes m + 2mR.
The dierent models were tested on R = 22 RIRs
taken from the database introduced in Section 4. Each
RIR was downsampled to fs = 800 Hz and truncated
to N = 4000 samples from the rst strong peak, selected as its starting point. The OBF-GMP pole grid
used G = 1000 poles with 20 dierent radii distributed
logarithmically from 0.75 to 0.99 and with 50 dierent angles placed linearly in the range [1, fs/2 − 1] Hz
(right plot of Figure 2). The error measure used to
compare the performance of dierent models with the
same number of parameters is the Normalized MeanSquare-Error (NMSE) in the time domain, averaged
over all RIRs, given by
Simulation Results
R
hNMSE = 10 log10
The simulations results presented here aim at comparing the OBF-GMP algorithm with OBF-MP and
the all-zero modeling. The obtained models are compared in terms of their ability to approximate a set of
1 X khr − ĥr k22
.
R r=1 khr k22
(7)
Figure 5 shows the average NMSE produced by the
OBF models using the two algorithms and by the all-
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zero modeling, for the same total number of model
parameters per RIR. It can be seen that the OBF
models provide a better approximation compared to
the all-zero model. Moreover, there is a signicant reduction in the approximation error when OBF-GMP
is used instead of OBF-MP, mainly resulting from the
use of a larger number of poles (about 30% more). The
fact that this improvement is not noticeable when the
number of parameters is small can be explained by
observing that OBF-MP tends to select poles closer
to the unit circle, which approximate better the main
strong resonances of the target magnitude response
with a small number of poles.
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Acknowledgement
[13]
This research work was carried out at the ESAT
Laboratory of KU Leuven, in the frame of (i) the
FP7-PEOPLE Marie Curie Initial Training Network
'Dereverberation and Reverberation of Audio, Music, and Speech (DREAMS)', funded by the European Commission under Grant Agreement no. 316969,
(ii) KU Leuven Research Council CoE PFV/10/002
(OPTEC), (iii) the Interuniversity Attractive Poles
Programme initiated by the Belgian Science Policy
Oce: IUAP P7/19 `Dynamical systems control and
optimization' (DYSCO) 2012-2017, (iv) KU Leuven
Impulsfonds IMP/14/037, (v) and was supported by
a Postdoctoral Fellowship of the Research Foundation Flanders (FWO-Vlaanderen). The authors would
like to thank Bang & Olufsen for the use of their
premises and equipment. The scientic responsibility
is assumed by its authors.
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[14] IEC 60268-13:1998, Sound system equipment - part
13: Listening tests on loudspeakers, 1998.
[15] A. Farina, Simultaneous measurement of impulse response and distortion with a swept-sine technique,
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[16] ISO 3382-1:2009, Acoustics - measurements of room
acoustic parameters - part 1: Performance spaces,
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[17] ISO 3382-2:2008, Acoustics - measurements of room
acoustic parameters - part 2: Reverberation time in
ordinary rooms, 2008.
[18] M. R. Schroeder, New method of measuring reverberation time,
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